The global holomorphic invariant αG(X) introduced by Tian [?], Tian and Yau [?] is closely related to the existence of Kähler-Einstein metrics. In his solution to the Calabi conjecture, Yau [?] proved the existence of a Kähler-Einstein metric on compact Kähler manifolds with nonpositive first Chern class. Kähler-Einstein metrics do not always exist in the case when the first Chern class is posi...

Two main problems face the construction of noncommutative actions for gravity with star products: the complex metric and finding an invariant measure. The only gauge groups that could be used with star products are the unitary groups. I propose an invariant gravitational action in D = 2n dimensions based on the constrained gauge group U(2) broken down to U(2) for n > 2. When n = 2 the gauge gro...

We prove that the family of all invariant sets of iterated systems of contractions R → R is a nowhere dense Fσ type subset in the space of the non-empty compact subsets of R equipped with the Hausdorff metric. An iterated function system (IFS for short) is a finite collection (T1, . . . , Tn) of weak contractions of a metric space X. By a weak contraction we mean a mapping T : X → X such that d...

We study the dynamics of magnetic flows on Heisenberg groups, investigating extent to which properties underlying Riemannian geometry are reflected in flow. Much analysis, including a calculation Mañé critical value, is carried out for $$(2n+1)$$ -dimensional groups endowed with any left invariant metric and exact, left-invariant field. In three-dimensional case, we obtain complete analysis lef...

A left invariant metric on a nilpotent Lie group is called minimal, if it minimizes the norm of the Ricci tensor among all left invariant metrics with the same scalar curvature. Such metrics are unique up to isometry and scaling and the groups admitting a minimal metric are precisely the nilradicals of (standard) Einstein solvmanifolds. If N is endowed with an invariant symplectic, complex or h...

We explicitly construct the metric and torsion couplings of two-dimensional (4,0)-supersymmetric sigma models with target space a four-manifold that are invariant under a U(1) symmetry generated by a tri-holomorphic Killing vector field that leaves in addition the torsion invariant. We show that the metric couplings arise from magnetic monopoles on the three-sphere which is the space of orbits ...

We study almost contact metric structures on 5-dimensional nilpotent Lie algebras and investigate the class of left invariant almost contact metric structures on corresponding Lie groups. We determine certain classes that a five-dimensional nilpotent Lie group can not be equipped with.

We show that a group G acts properly and effectively on a locally compact and σ-compact metric space (X, d) if and only if there exists a compatible G-invariant Heine-Borel metric dp on X such that G is homeomorphic to a closed subgroup of the group of isometries Iso(X, dp).